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I have calculated the exciton wave functions using the GW-BSE method:

$$\tag{1}|S\rangle = \sum_{cv\mathbf{k}} A_{cv\mathbf{k}} |cv\mathbf{k}\rangle$$

where $c$ for conduction and $v$ for valence states. Now, I would like to calculate the exciton center-of-mass group velocity $\mathbf{v}_S$ in the limit of $\mathbf{Q}\rightarrow \mathbf{0}$ ($\mathbf{Q}$ the center-of-mass momentum). As for a single electron-hole pair, the exciton velocity can be approximated as the average of the electron and hole velocities. I'm wondering if there exists similar relation like $\mathbf{v}_S \sim \sum_{cv\mathbf{k}} |A_{cv\mathbf{k}}|^2 ( \mathbf{v}_{c\mathbf{k}} + \mathbf{v}_{v\mathbf{k}} )/2$?

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Not a direct answer to your proposed approximation to the exciton group velocity as $\mathbf{Q}\to0$, but just wanted to point out that the latest version of Yambo supports the calculation of exciton dispersions, from which you can then calculate the exciton velocity at any $\mathbf{Q}$. Here are the details: Yambo 5: exciton dispersion.

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  • $\begingroup$ Thanks ProfM. The full exciton dispersion from Yambo is calculated based on Fourier interpolation. Can you comment on the accuracy of the interpolation? From my experience, it seems that the interpolation tends to make the curve at Q=0 quite flat. So I get nearly zero velocities. $\endgroup$
    – Xiaoming Wang
    Mar 3, 2021 at 9:19
  • $\begingroup$ If the dispersion has a minimum at Q=0, it should have a zero velocity. Why are you specifically interested in the velocity at Q=0 rather than elsewhere? $\endgroup$
    – ProfM
    Mar 3, 2021 at 12:29
  • $\begingroup$ I'm trying to calculate the optical activity which has a term proportional to the velocity at Q=0 (J. Phys. Soc. Jpn. 34, pp. 763-768 (1973) ). For 2D materials (Phys. Rev. Lett. 115, 176801) or materials with strong rashba effect, the velocity may be not zero. $\endgroup$
    – Xiaoming Wang
    Mar 3, 2021 at 17:03
  • $\begingroup$ The optical activity is related to the dielectric tensor. You can obtain it from the electric (and eventually magnetic, for some effects) excitonic dipoles. As for 2D materials and Yambo the dispersion is computed on a regular q-mesh. Then eventually interpolated. You can obtain the non analytic contribution including the long range term in the exchange. Using a dense k mesh you should be able to get the proper dispersion and then the group velocity. However, I do not see how this relates to the optical activity. $\endgroup$
    – Davide Sangalli
    Jul 10, 2022 at 21:01

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